Noting that the largest eigenvalue of a random matrix and the maximal increasing subsequence of a random permutation are both expressed by the Tracy-Widom distribution, I might broaden your question to appearances of the law of iterated logarithm in connection with Tracy-Widom. The application I am aware of involves the self-interacting one-dimensional stochastic process called "true self-repelling motion". The law of iterated logarithm of the position $X_t$ at time $t$ for this process was derived by Laure Dumaz in Large deviations and path properties of the true self-repelling motion: almost surely,
$$\lim {\rm sup}_{t\rightarrow\infty}\frac{X_t}{t^{2/3}(\ln\ln t)^{1/3}}=(2\kappa)^{-1/3},$$
with $\kappa=(2/27)|a_1|^3$ the first (negative) zero of the derivative of the Airy function.